prove geometrically that cos(x+y) = cosx cosy - sinx siny | 💯 guaranteed question | class : 11

Summary

In this video, the presenter teaches the derivation of the cos(x + y) trigonometric identity, a guaranteed exam question for class 11 students. The process begins with drawing a unit circle, dividing it into four parts, and marking key points. The derivation involves using trigonometric identities, geometric analysis, and properties of congruent triangles to arrive at the final equation: cos(x + y) = cos(x)cos(y) - sin(x)sin(y). The video emphasizes the importance of practicing the steps multiple times for better understanding and retention, with additional resources like a PDF available for further review.

Takeaways

• 😀 This video focuses on deriving the cos(x + y) identity for Class 11, which is an important and guaranteed exam question.
• 😀 The first step in the derivation involves drawing a unit circle and dividing it into four equal parts.
• 😀 Label the X and Y axes and mark four points on the circle: A, B, C, and D.
• 😀 The coordinates for the points are: A (1, 0), B (cos x, sin x), C (cos(x + y), sin(x + y)), and D (cos y, -sin y).
• 😀 Draw lines connecting points B and D, as well as B and O, and D and O to form the basic diagram.
• 😀 Use the angles formed by the points, such as angle AOB = x, angle AOC = x + y, and angle AOD = -y, to establish geometric relationships.
• 😀 The diagram helps in visualizing the proof, and understanding it is crucial for correctly solving the problem.
• 😀 The two congruent triangles BOD and AOC are used to prove relationships between the sides and angles of the unit circle.
• 😀 Apply the distance formula to show that the square of the distances BD and AC are equal, leading to the simplification of terms.
• 😀 The algebraic steps involve using the expansion of (a - b)^2 and (a + b)^2 formulas to eventually derive the final identity: cos(x + y) = cos x * cos y - sin x * sin y.
• 😀 Practice is emphasized as the key to mastering this derivation, with encouragement to repeat the process multiple times for full understanding.

Q & A

What is the identity being proved in this derivation?

The identity being proved is the cosine addition formula: cos(x + y) = cos(x) * cos(y) - sin(x) * sin(y).

What is the first step in the derivation process?

The first step is to draw a unit circle and divide it into four equal parts.

What are the four points marked on the circle?

The four points marked on the circle are: A(1, 0), B(cos(x), sin(x)), C(cos(x + y), sin(x + y)), and D(cos(y), -sin(y)).

What do the angles AOB, AOC, and AOD represent?

The angles are as follows: AOB = x, AOC = x + y, and AOD = -y.

What is the purpose of drawing triangles BOD and AOC?

The purpose of drawing triangles BOD and AOC is to use triangle congruence to show that the lengths of diagonals BD and AC are equal.

How are the lengths of diagonals BD and AC related?

The lengths of diagonals BD and AC are shown to be equal by the congruence of triangles BOD and AOC, which implies BD = AC.

What is the significance of squaring both diagonals BD and AC?

Squaring both diagonals allows us to use the distance formula and apply trigonometric identities to simplify and solve the equation.

What trigonometric identity is used to simplify the equation?

The Pythagorean identity sin²(θ) + cos²(θ) = 1 is used to simplify both sides of the equation.

What is the final result after simplification?

After simplification, the final result is cos(x + y) = cos(x) * cos(y) - sin(x) * sin(y).

What advice does the speaker give to students learning this derivation?

The speaker advises students to practice writing the derivation multiple times to fully understand the steps and make the process easier.